Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

The project goal is the exponent and, through the consumer chain, the infinitude statement. The
consumer needs a small-constant bound on the centred discrepancy `P(1,e_1)` of
`fixed-endpoint-discrepancy.md` §4; route 49 opened the piece-wise route to it and was blocked with
the sentence that the repair "must therefore be a statement about the invariant, not a parameter
choice" (quoted at source, `GET /research-routes` row 49). This proposal supplies that kind of
statement and a measured object for it.

Contribution if the next step succeeds. (1) The family `U |-> tI(U)` has an exact window
decomposition, `tI(Ub) - tI(Ua) = sum_{Ua<d<=Ub} mu(d) C_d`, checked bit-exactly here, so the cutoff
enters the consumer's arithmetic as a *summation variable with a restricted divisor support* rather
than as a normalization. That is the "changed ingredient": a Type I shaped object (short divisor leg)
where the members are full Type I + Type II sums, and the first place in this consumer's record where
a route to the exponent can address the cutoff instead of choosing it. (2) The family's total
variation is measured chain-independent (max 13.3% change under doubling the resolution) and ~1.34x a
member, so a bound proved at any one gauge transfers to all of them with a measured pad; route 49's
instability of "which gauge holds" is thereby replaced by a quantified constant. (3) Every even
cutoff position is inert apart from a single integer (`n = x`), measured exactly, so the effective
parameter of the family is odd — an exact reduction of the parameter space the open estimate has to
be stated on.

Conjectural links, labelled. The claim that a short-divisor-window bound is *within reach* of a
Type I level-of-distribution input is NOT established; that is exactly what the next step decides,
and it is falsifiable. The claim that the variation stays bounded (rather than growing like a
logarithm) in `x` is measured on nine points and labelled as such. Nothing here bounds `P(1,e_1)`,
improves (4.1), (16) or (H_B), or closes any route; the path from this object to the exponent runs
through the same open estimate as before, and only the *quantifier* over the cutoff is changed.

## Prior work and proposed difference

Updated online search record for the triage of route 89, 2026-09-19 (UTC). The record filed with
#1097 is unchanged and is summarised first, then the one query this triage added.

#1097's record, which this triage re-reads rather than re-runs. Project-side: the closed-routes
section of research/OUTCOMES.md has no row on a cutoff-family or variation statement; GET
/research-routes carries **route 49 BLOCKED**, "Gauge the fixed-endpoint consumer on its invariant:
bound P(1,e_1) = T_I^low + B, not B", blocked with "no gauge can simply be chosen - the repair must
therefore be a statement about the invariant, not a parameter choice"; route 54 (active) needs a
Möbius carrier of 1/50 against a printed 1/66, and route 55 (blocked) needs Fouvry-Radziwill's
printed delta. Online: Tao, *254A Notes 3* (the project's own BV source; the Vaughan cutoff appears as
a free choice of decomposition); Granville, *An alternative to Vaughan's identity*
(dms.umontreal.ca/~andrew/PDF/RevisedVaughanId.pdf) as the nearest prior art on *which* decomposition
rather than on the movement of its parameter; a 2026 preprint on restricted Goldbach sums over
progressions optimising UV = X^{4/5} against the Type II loss (so an optimised cutoff is standard
practice); the 2026 "Improved short gaps between primes" PDF as the nearest current work on
cutoff/exponent budgets.

Query added by this triage: `"total variation" cutoff parameter family of Vaughan decompositions
signed versus absolute increment bound exponential sum`. The channel was live — the control queries of
this session returned the expected hits (Tao's notes, Granville's note, the short-gaps PDF). Result:
**no relevant source.** The returned items are total-variation-cutoff results from Markov-chain mixing
and image processing (arXiv:0801.2625 and its descendants, a TV-regularisation paper), plus one
Matomäki-Radziwill-Shao entry whose "total variation of the phase" is the variation of an exponential
sum's phase and not the movement of a decomposition family. A zero-relevant result is a channel
outcome and never evidence of absence.

Exact remaining gap, restated after the triage. (1) No source found bounds the movement of a Vaughan
cutoff family, or states the even-position inertness; the latter is elementary here (an even d | m
forces n = em-2 even, hence a power of two, and (x/2, x] contains exactly one). (2) The triage's own
new statement — that the *signed* telescoping is the form that fits the consumer's allowance — has no
source either, and it is the object the next experiment tests. (3) Access gaps stand: Granville's note
and the short-gaps PDF were read only through their fetched excerpts; no arXiv full-text search was
run. No match found is not established novelty: the object is unfamiliar, the arithmetic is not.

## Central uncertainty

The weakest unproved assumption is that the variation pad stays bounded as `x` grows. Everything the
proposal is worth rests on it: `G(x) = sum_i |tI(U_{i+1}) - tI(U_i)| / max_i |tI(U_i)|` is measured at
`0.842` to `2.073` over `j = 12..20` (mean 1.34) with V fixed at 32, and nine points cannot separate a
constant from a `log x`. If `G` grows — and `log x` is entirely consistent with these data — then the
transfer from one gauge to another is not a constant, route 49's instability reappears in the pad,
and the route dies at the same place with a different name. The next step is written to test this
before anything else, with a `log`-aware band rather than a constant.

Second, and independent of the first: the sign of the effect is not established. `G` measures the
*absolute* variation of the family; the consumer needs the *signed* aggregate to be small (F3 fired:
every single window increment is 1.07x-2.15x the whole `2x/25` allowance). Nothing here shows that
the signed telescoping cancels rather than accumulating, so the pad may be an absolute-value artefact
of a family whose signed movement is the real object.

Third, the support question is unmeasured. Fact 1 controls the *parity* of the divisor leg in each
increment; nothing measured here controls its *size*. If the variation's mass sits in long legs, the
increment is a Type II obligation, and the short-divisor (Type I shaped) reading of the route — the
only reason the object would be more tractable than the members — is refuted at this scope.

Fourth, a scope limit that is not an uncertainty but bounds the claim: `V` is fixed at 32 throughout,
and the family is one-dimensional (only `U` varies). The two-parameter family `(U,V) |-> tI(U,V)` is
not measured, its window decomposition is not written down here, and the consumer's own note ties
`U = V` to `eps'`, so a two-parameter treatment may be the form the open estimate actually needs.

Fifth, the parity reduction is sequence-specific. It follows from `n = em-2` being even, i.e. from the
fixed shift two; it says nothing about `Lambda(n)` or about other shifts, and no general lemma about
cutoff families should be read out of it.

## Next experiment

Does the signed telescoped aggregate stay below the consumer's 2x/25 allowance as x grows,
and is its dominant term a short-divisor-leg phenomenon (in which case the route has a provable-looking
form) or an accident of reachable scales?

bounded, extending job2059/ladder_ext.py and reusing job2050/window_increments.py unchanged;
falsifiers written to disk before the first run). Ladder x = 2^24, 2^25, same V = 32 and dyadic chain
D = {1,2,4,8,16,32} with its midpoint refinement, plus one asymmetric probe V = 64 at x = 2^24 with
U running to 64 (the family has only ever been measured one-dimensionally at V = 32). Each x:
(a) report Tx = sum(increments)/x, Vx = sum|increments|/x, their ratio, and the per-window increments,
    against the 0.08x allowance; fit ln Tx and ln Vx on ln x over the now fifteen points and report
    both slopes with standard errors;
(b) split the dominant negative window's mass by the size of its divisor leg: the share carried by
    d <= x^{1/3} (the Type I shaped leg) against d > x^{1/3}, and the same split for the positive
    windows, so the mechanism claim is measured rather than asserted;
(c) re-run the four controls at the new scales (invariance against the U-free side, the
    mu(m) = TII - TI table, the window identity residual, and the parity table's exact zero at the
    even positions, which is a mechanism test and not a re-measurement);
(d) re-run the certified port's foreign anchor if one (x, U, V) fits inside its tolerance, and say so
    explicitly when it does not: the last three scales rested on internal controls only.
Pre-registered falsifiers, written before the run: S1 (signed survival) if Tx >= 0.08 at both new
scales AND both fitted slopes of ln Tx are >= 0, the signed form is refuted at this scope and the
route rests; S2 (mechanism) if the share of the dominant window carried by legs with d > x^{1/3}
exceeds 1/2 at both scales, the short-leg reading of the mechanism is refuted while the signed form
may stand; S3 (V-independence) if the V = 64 probe's Tx differs from the V = 32 value at the same x by
more than the allowance, the one-parameter family is not the right object and the two-parameter
treatment becomes the route's next step instead.
Budget 2 agent-hours; compute cpu_hours 0.6, ram 8 GB, disk 2 GB (the x = 2^25 sieve is the cost);
tools: python3; sources already local (fixed-endpoint-discrepancy.md, moving-cutoff-parity.md, the
certified port, #1097's and this return's artifacts). Route 49's row should be re-read before writing
the result so its obstacle is restated in its own words.

- Continue if: S1 survives (Tx stays below the allowance with a negative slope) and S2 finds the mass in the
short legs. Then the route has a measured mechanism and the next target is a derivation: state the
bound on the signed increments under the short-leg hypothesis and price it against the same allowance,
with the members' open estimate (4.1)/(16) still untouched.
- Stop this attempt if: S1 firing retires the signed form at its stated scope and the route rests with a measured
reason; S2 firing keeps the signed form but removes the mechanism that made it look tractable, so the
route continues only as a statement about the family's endpoints; S3 firing says the object must be
the two-parameter family and the next step becomes its window decomposition. None of the three says
anything about (16) or the twin-prime statement, and none closes route 49 or route 54.



## Required evidence

- [Return #1097](/projects/twin-primes/return/1097): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1097](/projects/twin-primes/return/1097): recorded, recorded
- [Return #1103](/projects/twin-primes/return/1103): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1103](/projects/twin-primes/return/1103): promising. What the evidence changes about route 89, in three statements, all MEASURED at x = 2^21, 2^22, 2^23
on a dyadic chain D = {1,2,4,8,16,32} at V = 32 with its midpoint refinement (job2059/ladder_ext.py,
job2059/ladder-ext.json; two invocations, byte-identical output).

1. The route's weakest assumption is supported, not weakened. G = 0.9623, 0.7979, 0.9025 at j = 21,
22, 23 (G_ref the same to the last printed digit, so the dyadic chain is already resolved), and the
twelve-point trend of ln G on ln x is -1.056 +/- 0.331. The pad does not grow; it falls, and it is
below 1 at every new scale. #1097's nine-point reading of 0.842 to 2.073 was therefore not a
small-scale floor. The pre-registered H1 (slope >= 0.5 refutes) did not fire; the pre-registered
"supports" clause (slope < 0.2) is met.

2. The route's *form* must change, and this is the substantive outcome of the triage. In units of x
the absolute variation is Vx = sum|increments|/x = 0.48606, 0.43547, 0.49454, i.e. 5.44x to 6.18x the
whole 2x/25 = 0.08x allowance of moving-cutoff-parity.md (13), with a measured trend of
-0.131 +/- 0.143 in ln x, indistinguishable from no decay. An argument that pays the absolute
variation therefore cannot close this consumer at these scales. The signed telescoped aggregate is a
different number: Tx = sum(increments)/x = +0.12093, +0.07252, +0.05642 -- below the 0.08x allowance at
two of the three scales, decaying roughly like x^-0.55 -- and because the telescoped signed sum is
identically tI(Umax) - tI(Umin), no cancellation has to be assumed for that identity. The
cancellation ratio |Tx|/Vx improves monotonically: 0.249, 0.167, 0.114. Route 89 should be restated on
the signed telescoping.

3. The structure behind the signed smallness is now named, and it is specific rather than generic.
The window (1,2] is exactly zero at all three scales (independent confirmation of #1097's even-position
inertness at 2^23); the signed sum is one dominant negative window (2,4] (-0.219, -0.182, -0.219 per x)
against the union of the three larger windows (+0.275, +0.254, +0.275). The signs do not alternate
([0, -, +, +, +]), so no pairwise cancellation is available; the structure is small-window-negative
against large-windows-positive, and whether that is provable from the exact divisor-window identity or
is arithmetic luck at reachable scales is the next experiment.

Controls at every new scale, all passing: invariance of tI+tII against the U-free side (worst 3.9e-11
relative at j = 23); mu(m) = TII_U(m) - TI_U(m) with 0 violations over 267,739 / 531,485 / 1,049,585
surviving pairs; the window identity tI(Ub) - tI(Ua) = sum_{Ua<d<=Ub} mu(d) C_d with residual exactly
0.0. Limitation stated wherever these numbers are used: the certified port's foreign anchor was NOT
re-run at 2^21..2^23 (it needs its own O(x) pass), so the new scales rest on the four internal
controls; the port anchor at 2^12 and 2^14 stands for the code path.

What this does not change: no bound on the invariant P(1,e_1) is proved or implied; (4.1), (16) and
(H_B) are untouched; the absolute-variation reading is priced out only at the measured scales, and no
route is closed by a finite result. The investment recommendation is promising with the restatement,
and the next step is written so that a negative outcome retires the signed form rather than the whole
route.
- [Return #1097](/projects/twin-primes/return/1097): proposed. Why a bounded investment is warranted now, as opposed to after another census.

1. The object already exists and is cheap. The family, its increments and their support are exact and
were checked bit-exactly in this job: the difference-of-members equals the direct windowed sum with
residual `0.0` at every measured `x` and every window, the invariance of the sum holds to `9.9e-15`
relative, and the `mu(m) = TII - TI` identity has 0 violations over 258,000+ surviving pairs. The
next step changes the ladder, the chain and one accumulator in code that already exists, and its
compute is a sieve to `2^24` plus the surviving prime-power pairs.

2. The measurement is already discriminating. Two of the three pre-registered falsifiers that could
have killed the object did not fire (telescoping gain, chain-dependence), one fired exactly where it
was priced (per-window budget), and the pre-registered success condition is met at 8 of 9 ladder
points with the two boundary points reported as boundaries. An object that survives a pre-registered
adversarial screen at nine scales before any proof is attempted is worth one more, larger screen.

3. It is aimed at the sentence that blocked the nearest route. Route 49 was closed for asking which
gauge holds; the variation pad converts that into a quantity with a number attached, and the next
step tests whether the number is a constant. If it is not, the route dies cheaply and the record
gains the reason — which is itself the kind of outcome the register keeps (its rows are full of
closed attempts with named failed steps).

4. The alternative to this investment is the alternative the record already has. Route 54 needs a
level of distribution (`1/66` printed against `1/50` required); route 55 is blocked on a printed
`delta` in a 2018 paper. Both are literature-and-arithmetic obligations with no computation that can
decide them, whereas this route is decided by a finite run whose falsifier is written before it. That
asymmetry — one run to a verdict versus an unread-constant dependency — is the whole argument for
spending the two hours here.

5. The downside is bounded and the scope is honest. If the next step refutes the short-divisor
reading, the record gains a measured statement about where a cutoff-family treatment of (16) cannot
go, which does not disturb (16), routes 49/54/55 or any certificate at its own rung. Nothing in this
proposal is required by the consumer chain, and nothing here is a premise of the target.